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A function is said to be homogeneous of degree $n$ if $f(\gamma x, \gamma y)=\gamma^{n} f(x, y)$ Similarly, a function of three variables is homogenous of degree $n$ if $f(\gamma x, \gamma y, \gamma z)=\gamma^{n} f(x, y, z) .$ Determine which of the following functions is homogenous, and if it is, give its degree.$$f(x, y, z)=3 x y^{2} z^{3}$$

6

Calculus 3

Chapter 6

An Introduction to Functions of Several Variables

Section 2

Partial Derivatives

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University of Michigan - Ann Arbor

Lectures

12:15

In calculus, partial derivatives are derivatives of a function with respect to one or more of its arguments, where the other arguments are treated as constants. Partial derivatives contrast with total derivatives, which are derivatives of the total function with respect to all of its arguments.

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A function is said to be h…

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Determine whether the func…

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A function $f(x, y, z)$ is…

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The function $f$ is homoge…

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A function $f$ is homogene…

Determine if each function…

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A function $f(x, y)$ is sa…

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So if we want to determine if this is going to be a homogeneous degree and we could go ahead and, um, just start on the left sides equation and see if we can get the right side to spit out will be f of camera X Gamel y Camas e and then we just plug it in over here. So be three times gamma X times, camel y squared Campbell or my wife z cute. And if we were to go ahead and distribute the squares, everything, and then multiply everything together that would give us so gamma to the six times three x y square to see a cute and now we'll be have right here is exactly what we started with. So this is equal to Gamma to the six times f of X y Z. And so this implies it is homogeneous, and it is of degree six. Let me do that of degree six

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